A New One-Twelfth Step Continuous Block Method for the Solution of Modeled Problems of Ordinary Differential Equations

Areo, E. A.; Omojola, M. T.

doi:10.4236/ajcm.2015.54039

Cited by 6 publications

(3 citation statements)

References 4 publications

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“…According to Areo and Omojola [21], the linear hybrid multistep method is said to be consistent if all the following four conditions are satisfied i the order of the method must be greater than or equal to one i.e. (p ≥ 1).…”

Section: Consistency Of the New Methodsmentioning

confidence: 99%

A New Uniform Fourth Order One-Third Step Continuous Block Method for the Direct Solutions of y′′ = f (x, y, y′)

Areo

Rufai

2016

BJMCS

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In this study, we applied the approach of collocation and interpolation to develop a new fourth order continuous one-third hybrid block method for the solutions of general second order initial value problems of ordinary differential equations. Three discrete schemes were derived from the continuous schemes. The discrete method was analyzed based on the properties of linear multistep methods and the method is found to be zero-stable, consistent and convergent. We reported an improved performance of the new method over the existing methods in the literature by solving four numerical examples and the approximate solutions obtained confirmed the superiority of our new developed scheme when compared with some latest existing approaches.

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Section: Consistency Of the New Methodsmentioning

confidence: 99%

A New Uniform Fourth Order One-Third Step Continuous Block Method for the Direct Solutions of y′′ = f (x, y, y′)

Areo

Rufai

2016

BJMCS

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“…The RAS of the developed schemes is considered in the light of Lambert [23]; Areo and Omojola [24]; Ibijola, et al [25].…”

Section: Region Of Absolute Stabilitymentioning

confidence: 99%

An Order Four Numerical Scheme for Fourth-Order Initial Value Problems Using Lucas Polynomial with Application in Ship Dynamics

Ukpebor

Omole

Adoghe

2020

International Journal of Mathematical Research

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From the time immemorial, researchers have been beaming their search lights round the numerical solution of ordinary differential equation of initial value problems. This was as a result of its large applications in the area of Sciences, Engineering, Medicine, Control System, Electrical Electronics Engineering, Modeled Equations of Higher order, Thin flow, Fluid Mechanics just to mention few. There are a lot of differential equations which do not have theoretical solution; hence the use of numerical solution is very imperative. This paper presents the derivation, analysis and implementation of a class of new numerical schemes using Lucas polynomial as the approximate solution for direct solution of fourth order ODEs. The new schemes will bridge the gaps of the conventional methods such as reduction of order, Runge-kutta's and Euler's methods which has been reported to have a lot of setbacks. The schemes are chosen at the integration interval of seven-step being a perfection interval. The even grid-points are interpolated while the odd grid-points are collocated. The discrete scheme, additional schemes and derivatives are combined together in block mode for the solution of fourth order problems including special, linear as well as application problems from Ship Dynamics. The analysis of the schemes shows that the schemes are Reliable, P-stable and Efficient. The basic properties of the schemes were examined. Numerical results were presented to demonstrate the accuracy, the convergence rate and the speed advantage of the schemes. The schemes perform better in terms of accuracy when compared with other methods in the literature. Contribution/Originality: The study uses Lucas polynomial for the derivation of a new class of numerical schemes. The schemes were implemented in block mode for approximating fourth order ODEs directly without reduction. It solves variety of problems including problem in Electrical Engineering. The schemes performs excellently better than other schemes in the literatures.

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“…Because of sufficient stability properties in hybrid block methods as a result of fixed step discretization, ability to use smaller step-sizes for approximations without error growth resulting from perturbation, they have been widely used for solving third order ordinary differential equations (ODEs) (see, Duromola (2019Duromola ( ), modebei et al, (2021, Haweel et al, (2021), Lawal et al, (2018), Aigbiremhon et al, (2021) and Obarhua & Kayode (2016), among others. Also, hybrid numerical methods have been used for the approximations of higher order ODEs (see, Abolarin et al, (2020) and Areo & Omole (2015)) and results from their applications showed improved absolute errors. Similarly, adaptive polynomial method was formulated and implemented on linear, non-linear and a thin film flow problem and compared with some other hybrid block methods (see, Momoniat andMahomed (2010), Mecheel et al, (2013) and Yap et al, (2014)).…”

Section: Introductionmentioning

confidence: 99%

Selected Single-Step Hybrid Block Formula for Solving Third Order Ordinary Differential Equations With Application in Thin Film Flow

Atabo¹,

ARGAWAL²,

KWALA³

et al. 2023

FJS

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This research paper examines the derivation of selected hybrid single-step block method for the numerical integration of third order ordinary differential equations. The method has the advantage of selecting only odd off-grid points within the interval of interest. The basis function for the formula is interpolated at three selected non-grid points within a single-step interval and collocated at all points. Further analysis of the basic numerical properties were established. The method was found to be A-stable, zero-stable and consistent. The small scale errors observed from numerical experiment indicate that the derived formula has better numerical approximations than some comparable methods in literature, while its application on practical thin film flow problem also showed improved solutions.

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A New One-Twelfth Step Continuous Block Method for the Solution of Modeled Problems of Ordinary Differential Equations

Cited by 6 publications

References 4 publications

A New Uniform Fourth Order One-Third Step Continuous Block Method for the Direct Solutions of y′′ = f (x, y, y′)

A New Uniform Fourth Order One-Third Step Continuous Block Method for the Direct Solutions of y′′ = f (x, y, y′)

An Order Four Numerical Scheme for Fourth-Order Initial Value Problems Using Lucas Polynomial with Application in Ship Dynamics

Selected Single-Step Hybrid Block Formula for Solving Third Order Ordinary Differential Equations With Application in Thin Film Flow

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